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<p>We consider the Dirichlet-to-Neumann operator <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2796_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation> associated with an elliptic operator <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2796_Article_Equ1.gif" Format="GIF" Height="54" Rendition="HTML" Resolution="72" Type="Linedraw" Width="166" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} \mathcal {A}u = - \sum _{k,l=1}^d \partial _k (c_{kl}\, \partial _l u) \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi mathvariant="script">A</mi> <mi>u</mi> <mo>=</mo> <mo>-</mo> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>d</mi> </munderover> <msub> <mi>∂</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow> <mi mathvariant="italic">kl</mi> </mrow> </msub> <mspace width="0.166667em" /> <msub> <mi>∂</mi> <mi>l</mi> </msub> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with possibly complex coefficients. We present five interesting open problems related to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2796_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation>.</p>

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Five Open Problems on the Dirichlet-to-Neumann Operator with Variable Coefficients

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  • 发表日期：

    2025年6月4日

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  • A. F. M. ter Elst,
  • E. M. Ouhabaz

摘要

We consider the Dirichlet-to-Neumann operator \(\mathcal {N}\) N associated with an elliptic operator \(\begin{aligned} \mathcal {A}u = - \sum _{k,l=1}^d \partial _k (c_{kl}\, \partial _l u) \end{aligned}\) A u = - ∑ k , l = 1 d ∂ k ( c kl ∂ l u ) with possibly complex coefficients. We present five interesting open problems related to \(\mathcal {N}\) N .